« International Environmental

Cooperation : A New Eye on

the Greenhouse Gases Emissions’ Control »

Mélanie HEUGUES

DR n°2009-04

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International Environmental Cooperation: A New Eye on the Greenhouse Gases Emissions’ Control

Mélanie HEUGUES1

Abstract

We examine the formation of International Environmental Agreements (IEAs)

modelled as a two-stage non cooperative game. Following Barrett (1994), Finus

(2001) and Diamantoudi & al. (2006) and filling out their approach, we analyse

the level of cooperation that can be reach when countries’ strategies are

complementary. We find that when strategies exhibit weak complementarities,

the unique stable agreement can consist of half the countries involved in the

negotiation and thus, without any form of commitment, linkage or transfers

between countries. These results, established analytically, strongly contrast with

those of the previous authors and are a lot more optimistic. Nonetheless, even if

the incentives to free-ride are less strong, we do not observe the formation of the

“grand” coalition: not all the countries sign the agreement. We also provide

some results of comparative static. We analyse, for example, the level of

cooperation which only depends on the number of countries concerned with the

problem of climate change and on the perception they have of its seriousness.

Key Words: Non-cooperative game theory, International environmental

agreements, Strategic complementarities, Coalition formation

JEL Codes: D62, Q28

1 LAMETA, University of Montpellier. Mail: heugues@lameta.univ-montp1.fr

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1. Introduction

The problem of climate change belongs to the class of global environmental problems in the

sense that damages a country bears, do not depend on its own emissions but on aggregate

emissions: every country emits greenhouse gases into the atmosphere and every country will

be affected by the climate change caused by the accumulation of these gases. Even if the

impacts of global warming can be sustained differently by each country, environmental

changes only depend on world emissions. The main reasons why the control of greenhouse

gas emissions (afterwards GHG emissions) now covers an international dimension are the

following: (i) once emitted, GHG concentrations become quickly and rather uniform around

the world, (ii) as a consequence of trade globalization of these last 50 years, the sources of

pollution come from economic activities which are less and less geographically bounded and

(iii) national environmental policies are increasingly interdependent. Consequently, the

economic regulation of the environment has to be undertaken at an international level2.

Empirical observations from the last twenty years strongly support this claim. Taking into

account the reality of global warming and of the responsibility of human activities on climate

imbalances, some governments have begun environmental policies to curb their GHG

emissions. The Kyoto Protocol, in effect since 2005, is the most recent example of an

international coordination to take action in the matter of climate change.

However, the main source of failure of the international system is States’ sovereignty. In our

realm, there is no supranational authority to enforce arrangements between countries. When

environmental problems are global, involving all countries, each of them has to decide

voluntarily to provide or not the public good ‘Environment’, i.e to reduce its GHG emissions.

It also means that International Environmental Agreements (IEA) have to be self-enforcing in

the sense that they are immune to deviation by the countries involved. Theoretical models

developed until recently try to explain the emergence of cooperation between countries when

they are subject to strong environmental externalities. An important result of this literature is

that cooperation emerges even in a non-cooperative setting (Carraro & Siniscalco, 1991;

Barrett, 1994; Finus & Rundshagen, 1998; Diamantoudi & al., 2006 among others). These

results are based on non-cooperative game theory and more particularly on endogenous

coalitions’ formation theory with spillovers. Game theory is particularly appropriate to

2 See Barrett (2005) for an overview of the type of environmental problems IEAs are meant to address.

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account for strategic interdependence between agents. In our framework, the literature shows

that there exists a stable coalition, as defined in industrial organization by D’Aspremont & al.

(1983) and extended to IEA (Barrett, 1994; Bauer, 1992; Hoel, 1992). In other words, a subset

of countries involved in the negotiation find beneficial to jointly diminish their emissions,

meaning that it is in their self-interest; the agreement is self-enforcing. However the

equilibrium number of signatories is in general rather small with respect to the number of

countries concerned with the environmental problem. No more than four countries will form

the stable coalition without regard on the number of countries involved in the negotiation

(Diamantoudi et al., 2006). Hence, the impacts of the actions undertaken remain limited and

do not allow restoring the global optimum.

Actually it turns out that the robustness of this result on the emergence of environmental

cooperation without any trigger or stick mechanisms is based on one assumption which

appears in most of the studies on international environment but is seldom discussed: the

pattern of interdependence among countries. This latter is described by the slope of the best-

reply functions, and it is fundamental for understanding the effectiveness of cooperative and

non-cooperative emission control (Carraro and Siniscalco, 1991)3. Whereas they are derived

from the most general form of the payoff function used by the literature on IEAs, the results

discussed above suppose that countries’ strategy are substitutable4.

In this paper we are looking for what happens when strategies become complementary. This

assumption means that, if a country increases its emissions, the others will have a positive

incentive to do the same. This idea is meaningful if we look at the current economies of

developed counties and their political and trade interconnections. The formation of free trade

areas (regionalization) in Europe, Asia or America have led to a boom of trade between

countries belonging to these blocks. Obviously, if the United-States falls in a recessionary

period tomorrow, the level of activities of its trade partners will fall as well.

4 Carraro and Siniscalco (1991) already underlined this idea, but confined their study to the case where the slope of the best-reply functions are negative. They find that the more negative it is, the larger the incentive to deviate from any coalition. 4 For an overview of assumptions and results of game theoretical analyses of IEAs see Finus (2001 and 2003a).

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Going further into prior findings, we analyse in this paper the level of cooperation that can be

reach when countries’ strategies are complementary. The countries, which are the decision-

making units, are interdependent in the sense that their economic activities are more and more

connected and integrated. The study of coalition formation under strategic complementarities

has already provided some interesting results in considering the case of the spread of catalytic

converters in automobiles (Heal, 1993; Barrett, 2003). Under discrete policy choices and

multiplicity of equilibria, these authors find that the challenge for countries concerned with

the protection of the environment is not anymore how to sustain full cooperation? but rather

how to get to the preferred equilibrium? In other words, the existence of reinforcement effects

between countries’ strategies turns the initial preoccupation into a coordination problem.

Therefore, with a threshold constraint, an IEA may help to coordinate States’ behaviour such

that the welfare-superior equilibrium is sustained.

What distinguish our findings to previous is that, we do not consider perfect

complementarities between countries. In the tradition of the first models, the question always

is: how many countries will sign the agreement? Thereby, in this paper, we choose to consider

the problem of international pollution control in a similar way to Diamantoudi & al. (2006)

with quadratic benefit and damage functions. One main difference is the specific functional

forms which exhibits strategic complementarities. So, we study the incentives schemes of

sovereign countries which sign a treaty. The equilibrium number of countries participating in

an IEA is derived by applying the internal and external stability concept borrowed from the

oligopoly literature. We show that, under these assumptions, cooperation is largely greater.

Nonetheless, all the results depend on the number of countries involved in the negotiation and

their perception of environmental damages with respect to the benefits of their GHG

emissions. Moreover, even if an agreement will always be signed, it will never be the “grand

coalition”.

The paper is organised as follow. In section 2, the fundamental functions and assumptions of

the game are defined. In the third section, we derive the non-cooperative and the full

cooperative outcomes. These solutions are used as a benchmark to analyse signatories’ and

non-signatories’ behaviour. Finally, we establish analytically the size of the stable agreement,

i.e. the level of cooperation that can be reached by the countries under strategic

complementarities. The last section concludes. The proofs of all results are given in the

appendix.

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2. Fundamental functions and assumptions of the emission game

We consider n symmetric countries with N = {1, ..., n}. Linked to economic activities, each of

them emits greenhouse gases (GHGs), 0≥ix , that mix uniformly in the atmosphere. In its

most general form, the payoff function of country i, if , is expressed as the difference

between the benefits of emissions, )( ii xB , and the damages linked to global emissions,

∑ ∈ )( iNii xD . Then, if can be written:

∑−= j jiiii xDxBf )()( , Nji ∈∀ , .

Emissions are viewed as an output of the production and consumption of goods from which

benefits are derived, while the payoff function describes the welfare implication from

emissions at an aggregate level. Even if the impacts of global warming can be supported

differently by each country, environmental changes only depend on world emissions. Hence,

the damage function of the country under consideration does not depend on its own

contribution but rather on the sum of the emission levels supplied by all countries. So,

emissions in a country generate an externality causing environmental damage in this country

but also in the other countries.

Since the countries are assumed to be identical, we drop the subscripts from the functions.

Moreover, we will note x, y and z respectively the emission level of the country under

consideration, the aggregate emission level of the (n – 1) other countries, and the global

emission level, i.e., z = x + y.

We will assume that the benefit function is increasing in the level of emissions, i.e.

0)(' >ixB . Similarly, given that emissions disperse uniformly in the atmosphere, the

damage function is supposed to increase in the global emissions z, i.e. 0)(' >⋅D over R+ .

For each country, we assume hereafter the following specific benefit and damage functions: 2/1)1()( −= xbxB , with b a positive benefit parameter, and

2/1)()( yxcyxD +=+ , with c a positive cost parameter.

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In the current literature parameters b and c are usually introduce to analyse the relative impact

of cost of emissions compared to benefit. Parameter b is often interpreted as an opportunity

cost of abatement whereas in Endres (1997) parameter c is interpreted as the environmental

awareness of the society of a country. Particular attention will be given to the effect of a

change in these parameters on the success and stability of an IEA.

Each of these functions is increasing in the individual emission level but at a decreasing rate.

We assume a concave damage function because when damages only depend on aggregate

emissions, it’s the only way for payoff functions to exhibit strategic complementarities.

For a country, the form of the payoff function is then: 2/12/1 )()1(),( yxcxbyxf +−−= , with 1 ≤ x ≤ K,

With K accounting for a capacity constraint. It means that a country cannot produce infinitely

GHG’s emissions or that its economic activities are bounded. Similarly, notice that our

function set forth a lower bound on emissions in the sense that a country cannot reduce its

economic activities under some level.

In the following, each country is supposed to maximise its payoff function to determine its

level of emissions. We first consider the non-cooperative and the full cooperative cases. They

are next used as a benchmark to study the endogenous formation of an environmental

agreement.

3. The non-cooperative and full cooperative outcomes

3.1. The pure non cooperative case

In the pure non-cooperative case each country chooses its emission level taking the other

countries’ emission as given:

2/12/10 )()1(),( yxcxbyxfMax x +−−=≥ .

For the country under consideration the necessary condition leads to the following reaction

function:

1)( 2

2

−+

=γγ yyx with

bc

=γ.

In what follows, bc /=γ is defined as the cost/benefit ratio of GHG emissions. It

represents the perception that countries have of environmental damages in comparison with

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the benefits they obtain from their emissions. Given the assumption of identical countries,

xny )1( −= . The emission levels for a country and the global emissions in the non-

cooperative case are respectively:

n

xnc−

= 2

2

γγ

,

n

nnxz ncnc−

== 2

2

γγ

.

The constraint on the emission levels is observed for all n when 1

2−

>KnKγ .

Given equilibrium emission levels, the shape of the payoff function of a country in the non-

cooperative case is:

2/12

22/1

)()1(

nbnf nc

−−

=γ

γ

.

From our assumptions on the payoff functions and the solutions of the maximisation problem,

we can establish a preliminary result.

Result 1:

Under our assumptions, countries’ strategies exhibit weak complementarities. It follows that:

(i) Individual and aggregate equilibrium emissions levels, ncx and ncz are increasing

in n and decreasing in γ;

(ii) The equilibrium payoff of each country is decreasing in n and increasing in γ until 2/1)12( −= nγ ; beyond it decreases in γ.

When countries’ strategies exhibit complementarities it means that, if a country increases its

levels of production and consumption and thus its level of emissions, the others have a

positive incentive to do the same. Hence, the marginal utility a country gets from its emissions

is increasing in the level of emissions of the other countries.

Moreover, when considering countries as trade partners, the larger is n, the more a country

will increase its trade relationships and thus its economic activities. However it also means

that the level of emissions of each country will be increasing in the number of countries

involved. As a result of the non-cooperative case, the larger is the number of countries

concerned with the environmental problem, the greater the individual and aggregate

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equilibrium emissions levels and the higher the marginal utility a country derived from its

emissions. This result is immediately linked to strategic complementarities between countries.

Furthermore, there is nothing new in saying that the more the damages are perceived seriously

compared to the benefits of emissions, the lower the equilibrium emission level of a country.

In other words, the marginal utility of emissions of each country is decreasing in the

cost/benefit ratio. Therefore, referring to the second point of result 1, as soon as the

perception of damages is not too strong ( γγ < ), equilibrium payoffs are increasing in the

perception of damages. The existence of γ restrains the emission level of each country and

then the importance of externalities. The larger is γ , the more the levels of emissions are

contained and the higher the equilibrium payoffs. Beyond the threshold γ , what happens is

that damages are perceived so seriously that all countries emit as less as possible and cannot

reduce their emission level as much as in the prior case. Hence, the abatement effort does not

suffice to balance the importance of the externalities. Yet countries have no choice except to

suffer the losses linked to environmental damages without having a real impact on it.

Finally the last point is linked to the fact that the game is a game with negative externality,

and countries, when choosing their emission level, do not take into account the impact of their

choice on the other countries. This phenomenon is amplified by the fact that countries’

strategies are complementary: if a country increases its emissions, the others have an

incentive to increase their own emissions. So, at the Nash equilibrium there is an increasing

incentive to have greater emissions. Thus, the larger is n, the greater are the environmental

externalities and the lower are the equilibrium utilities. In the same way, the larger is n, the

higher will be the global emissions and the more severe the environmental problem. In this

setting, countries should be more incline to cooperate. Before studying endogenous

cooperation between countries, we now present what happens in the full cooperative case. In

other words, we analyse the situation when each country takes into account the negative

externality it imposes upon other countries.

3.2. The full cooperative case

In the full cooperative case, we assume that the whole countries maximize their joint payoff.

Formally we solve the following maximization problem:

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⎟⎠⎞

⎜⎝⎛ −−≥

2/12/10 )1( cz

nzbnMax z

.

The necessary condition leads to the following aggregate emission level of full cooperation:

12

22

−=

γγ

nnz c

.

Since each country contributes the same fraction of total emissions, the per-country emission

level is:

12

2

−==

γγ

nn

nzx c

c.

Given equilibrium emission levels, the shape of the payoff function for each country in the

full cooperative case is:

2/12 )1( −−= γnbfc .

As in the non-cooperative case, we establish the same kind of comparative static results when

all countries cooperate (result 2) and then we compare the equilibrium emission levels and

equilibrium payoffs with the non-cooperative outcomes (result 3).

Result 2:

In the full cooperative case, we find that:

(i) Individual equilibrium emission level, cx is decreasing in n whereas aggregate

equilibrium emission level, cz is increasing in n;

(ii) Individual and aggregate equilibrium emission levels, cx and cz are decreasing in γ;

(iii) The equilibrium payoff of each country is decreasing in n and in γ.

Under our assumptions we find that when all countries cooperate, individual emission levels

are decreasing in n. This observation is linked to the fact that the game is of negative

externalities. When cooperating, countries take into account the negative impact of their

emissions on the other countries. The larger is n, the larger is the scope of emissions and the

lower will be the individual emissions. In other words, countries have to take into account n

times the environmental damages linked with their emissions instead of one time.

Nonetheless, global emissions keep on increasing in the number of countries concerned with

the environmental problem. Consequently, equilibrium payoffs are decreasing in n. However,

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contrary to the non-cooperative case, it is because abatement efforts are increasing in the

number of countries concerned with the environmental problem.

As previously individual and aggregate emission levels are decreasing in the cost/benefit

ratio, i.e. the stronger is the perception of the seriousness of the environmental damages, the

more countries will reduce their individual emission level. The payoff of full cooperation is

decreasing in γ because countries, in cooperating, do their maximum to reduce their impact on

the environment. The stronger is their perception of the damages compared with the benefits

of emissions, the lower is their utility.

Before contrasting emission levels and payoffs under full cooperation with the non-

cooperative outcome, we define the indexes I1 and I2. The first one corresponds to the relative

difference between global emissions in the Nash equilibrium and in the social optimum. It

defines, in relative terms, the ‘degree of externality’ (Finus, 2001). The latter index states the

relative difference between joint payoffs in the social optimum and in the Nash equilibrium5:

nc

cncz

zzI )(1

−=

,

c

ncc

c

nccnf

nfnff

ffI )()(2

−=

∑∑−∑=

.

Result 3:

We can assess the three following points:

(i) Individual and aggregate equilibrium emission levels under full cooperation, cx

and cz are such that ncc xx < and ncc zz < ;

(ii) Equilibrium payoffs are such that ncc ff > ;

(iii) I1 is increasing in n and decreasing in γ whereas we find the reverse result for I2.

We find the standard results of games exhibiting negative externalities: countries are better off

under full cooperation than in the non-cooperative setting. As each country takes into account

5 The second index in terms of payoffs can be defined because all payoffs for all countries and

in all cases are of the same sign.

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the negative impact of its emissions on the others, the country reduces its individual emission

with respect to the non-cooperative outcome. Globally, aggregate emissions also fall. Each

wins of such a strategy.

The last point means that, the larger the number of countries which suffer from the global

externality and the lower the cost/benefit ratio of emissions, the more important it is to reach a

cooperative agreement. The first result is characteristic of the provision of public goods. The

latter indicates that emission reduction is particularly attractive if the perception of damages is

not to strong with respect to the benefits of emissions. In interpreting parameter b as the

opportunity cost of abatement, the bigger is b relative to environmental costs (γ is low) the

higher will be the emissions in the non-cooperative outcome and the more inefficient will be

the situation from the global society’s point of view. In contrast, if environmental damages

are very high, only moderate emission reductions are advisable from a global perspective.

Cooperation is then less attractive because individual emissions are already as low as

possible. Hence, only moderate emission reductions can be reached.

Relatively to the second index, gains linked to cooperation are higher, the number of countries

concerned with the environmental problem is not too large and the stronger is the perception

of damages. In other words, linked to the cost/benefit ratio, cooperation will pay less if

opportunity costs of abatement are high or environmental awareness is low (γ is low). Hence,

the higher will be the ‘degree of externality’ the lower will be the relative difference in terms

of payoffs (individual actions of abatement are taken more easily when the ‘degree of

externality’ is increasing).

For any n and γ, the full cooperative outcome is the best possible cooperative solution.

However, in the realm of one stage game with simultaneous play, each country has an

incentive to take advantage of the agreement and to free ride on the emission reduction

achieved by the countries complying with the agreement. Thereby each country earns a lot

more in deviating unilaterally, i.e. it benefits of a better environment without paying the cost.

In what follows we consider the two-stage framework. In a first step each country decides to

be a signatory or a non-signatory. In a second step, they play the emission game. In doing so,

the incentive to free ride on the coalition’s cooperating efforts may be offset by the

adjustment of the coalition’s emissions upon a member’s deviation.

4. The endogenous coalition formation game

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In this section, we analyse the emergence of an IEA as the equilibrium of a two-stage game.

In the first stage, countries choose independently and simultaneously whether to join an

agreement or to remain a non-signatory. Given the option chosen in the first stage, signatories

and non-signatories then decide in a second stage of their emission levels. Hence we restrict

our attention to the formation of only one, non-trivial6, coalition. It’s assumed that countries

forming an IEA coordinate their policies by jointly maximising the aggregate welfare of the

coalition and given the behaviour of non-signatories. Given the assumption of symmetric

countries, we assume that signatories agree on a symmetric abatement scheme which implies

equal payoffs. In contrast, each non-signatory behaves non-cooperatively and chooses its

emission level independently, i.e., given the behaviour of signatories and the behaviour of the

other non-signatories. Let NS ⊂ the set of countries that sign an agreement and N\S the set of

those who do not. We note the size of the coalition, the emission level of a

signatory while the total emission generated by the coalition is . Similarly each non-

signatory country emits yielding a total emission level . Finally, we

assume that players choose their strategies in both stages simultaneously7.

The next two sub-sections solve the two-stage game by backward induction. The equilibrium

emission levels of the second stage are determined first, given the number of signatories, s.

4.1. Signatory and non-signatory behaviours in the emission game

Signatories are assumed to act collectively rather than independently. Particularly, they

choose the emission level that maximises their aggregate payoff. Notice that, in that case, the

damages are a function of both signatories’ emissions and non-signatories’ emissions.

Formally, signatories solve the following maximization problem:

( )2/12/10 ))(()1( nsssxs xsnsxcxbsMax −+−−≥ .

The necessary condition leads to the following best response function:

1)()( 2

22

−−+

=s

xsnsxx nsnss γ

γ

.

6 A non-trivial coalition is a coalition of at least two members. 7 This assumption is referred as the Nash-Cournot assumption and is in line with the works of

Carraro and Siniscalco, 1991; Bauer, 1992; Hoel, 1992. For an alternative assumption, see

Barrett 1994 and Finus, 2001.

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To determine completely the emission level of the coalition with respect to its size, we have to

solve the maximisation problem of a non-signatory.

A non-signatory choose its own emission level, xi, given the emission levels of the s

signatories and of the (n – s - 1) other non-signatories. Formally, a non-signatory solves the

following problem:

( )2/12/10 ))1(()1( inssixi xxsnsxcxbsMax +−−+−−≥ .

The necessary condition leads to the following best response function:

1)1(),( 2

2

−−−++

=γ

γ nssnssi

xsnsxxxx.

This reaction function is a function of both the emission levels of the signatories and of the

other non-signatories. By symmetry, all non-signatories will play the same equilibrium

emission level. So, we can establish the emission level of a non-signatory, xns, as a function of

the emission level of the coalition:

ins xsnx )( −= ,

snsxxx s

sns+−

+= 2

2)(

γγ

.

The intersection of both best-response functions leads to the following equilibrium emission

levels, respectively of a signatory and of a non-signatory:

ssnsnx s

−+−+=

)(1 22 γ ,

1)(1 2 −+−+=

snsnsx ns γ .

The aggregate emission level nssT xsnsxz )( −+= is:

1)( 2

2

−+−=

snsnszT γγ

.

For any n, s ≥ 1 and γ the equilibrium emission levels are positive and belong to the feasible

action set before-defined. To determine the welfare levels of signatories and non-signatories

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for any given s, we substitute the equilibrium emission levels sx , nsx and Tz into the

corresponding payoff functions. The shapes of the payoff functions, respectively of a

signatory and of a non-signatory, are:

2/122

22/1

])([)1(ssns

sbnf s−+−

−=

γγ

,

2/12

22/1

]1)([)1()(

−+−−

=sns

nsbfns γγ

.

From these last equilibrium outcomes, we can establish the following result:

Result 4:

The analysis of the outcomes of the second-stage emission game leads to the following points:

(i) For any s and n, emission level of a signatory will always be below the emission

level of a non-signatory, but both reduce their emissions compared with the non-

cooperative outcome;

(ii) The payoff of a signatory is always below the one of a non-signatory: nss ff < ,

but the difference is decreasing in the number of cooperating countries;

(iii) The larger is the coalition s, the lower the individual emission levels, sx and nsx ,

and so, the lower the global emissions Tz ;

(iv) Both type of payoff function are increasing in the number of signatories, s.

To understand this result, notice that we are in a mid-case between non-cooperation and full

cooperation. The full-cooperative and the pure non-cooperative outcomes are special cases of

the outcomes with endogenous formation of an agreement. That is, when s = n, the problem

reduces to the full-cooperative solution ( cs xx = and cs zz = ), while when s = 1, it

reduces to the pure non-cooperative solution ( ncs xx = and ncs zz = ).

The fact that a signatory pollutes less in term of GHG emissions than a non-signatory is not

new when cooperation is partial. Cooperation aims at this result. Though, and contrary to a

non-signatory, a country which signs the agreement only takes into account the negative

externality the others signatories bear because of its emissions. A more remarkable thing

relative to the non-cooperative outcome is that, the cooperation between a subgroup of

15

countries also leads the non-signatories to reduce their emission levels. The free-riding

phenomenon thus takes a different form as postulated before in the literature: instead of

canceling the efforts provided by the cooperating countries, a country which free-rides will

abate its emissions but less than a signatory. When payoffs exhibit strategic

complementarities, a signatory reduces more its GHG emissions than a non-signatory; and the

larger is the number of signatories, the higher the individual abatement efforts of both type of

country. Though, the larger is s, the lower the global emissions.

The second point is a consequence of the first one. All countries suffer the same damage:

global emissions are the same for all; but a signatory in reducing more its emissions has lower

benefits than a non-signatory. Consequently, a country which signs the agreement will always

earn less than a country which does not. We can also show that this difference in payoff is

decreasing with the size of the IEA. In other words, the larger the number of cooperating

countries, the lower is the relative difference between a signatory and a non-signatory payoff.

Contrary to the results established earlier in the literature, we find that signatories as non-

signatories reduce their GHG emissions. The effort of the coalition is not anymore (partly)

cancelled by the decisions of the non-cooperating countries. However, a non-signatory will

always have a higher welfare than a signatory. Under these conditions, the question is then

what is the size of the stable IEA?

4.2. The size of stable IEAs with strategic complementarities

To determine how many countries will sign the agreement and how many will not in the first

stage, we use the concept of internal and external stability. An IEA is said to be internally

stable if none of its members wants to leave; on the opposite, an IEA is externally stable if

none of the outsiders wants to join it. Formally, s* is an equilibrium participation level if it

satisfies:

)1*(*)( −≥ sfsf nss and )1*(*)( +≥ sfsf sns .

As Diamantoudi and al. (2006), we are interested in determining analytically the size of the

stable agreement. In equalizing )1( −sf ns and )(sf s , and allowing s to take non-integer

values, it provides an analytical solution s~ . But we are interested in determining the largest

16

integer s* ≤ s~ , such that it satisfies the internal and the external stability conditions (not

necessarily with equality).

Formally, we are looking for ss ~* ≤ such that 0)~()1~( =−− sfsf sns . We find the

following solution:

)12(2)1()52(~

2

24

−Δ++−++−

=γ

γγ mnns,

With ]2)43(3)[21(4)]1()52([ 242224 −++−−−+−++−=Δ nnnnm γγγγγ > 0.

The fifth result establishes the variations in the equilibrium number of signatories with respect

to the number of countries concerned with the environmental problem and the perception they

have of its seriousness.

Result 5:

Under our assumptions, there exists a unique stable IEA whose size only depends on n and γ:

(i) For any γ and n = 2, both countries are always better off in cooperating;

(ii) For n ≥ 3, if n→2γ (the most favourable condition), the stable coalition s* gets

together between 50% and 75% of the countries concerned with the environmental

problem; the larger is n, the most s* tends toward 50% of the countries.

(iii) For n ≥ 3, if n42 →γ (the perception of damages is very strong with respect to

the perception of benefits), the stable coalition brings together only two or three

countries.

The stable agreement is the larger when the perception of damages is as low as possible. It

also corresponds to the case in which countries pollute the most when there is no cooperation.

When n→2γ cooperation is required (I1 is the biggest) and is established. Cooperation

will always bring together more than half of the countries. Nonetheless it stays partial in the

sense that beyond s*, it pays more to be a non-signatory than a signatory. It becomes too

costly to reduce its emissions as much as the signatories.

Given n, the stronger the parameter γ, the lower will be the equilibrium number of signatories.

It can be explain by the fact that when policies are unilateral, the stronger is the perception of

damages relative to benefits of emissions, the lower will be the individual emission level.

Hence, cooperation is less attractive and the size of the stable agreement is smaller.

17

The last result establishes the environmental impact, in term of global emissions, of the

cooperation between a subgroup of countries:

Result 6:

Under our assumptions, we show the following points:

(i) As soon as s

sn 122 −

+<γ , the environmental impact of the coalition is increasing

in its size;

(ii) Beyond this threshold, the environmental impact of the coalition is decreasing in

its size.

If a subgroup of countries agrees to reduce their individual emissions, global emissions will

decrease relative to the non-cooperative situation and all countries will gain of such a thing. In

the most favourable case, cooperation can lead to a cut off of global emissions of 50%.

However, the perception of damages with respect to the benefits of emissions must be below

some level for the cooperation (even if partial) to have a real impact. If it is, we show that the

reduction of global emissions relative to the non-cooperative case is increasing in the size of

the coalition. When γ is beyond the threshold above-defined, the reverse result hold and the

environmental impact of the stable coalition decrease with the equilibrium number of

signatories.

Depending on the parameter γ, this last result illustrates the cases when cooperation is needed

and the cases when it is not. The institutions intended to address international problems then

also vary: “Full cooperation is almost always desirable but rarely essential to the supply of

environmental public goods” (Barrett, 2005).

5. Conclusions

The results obtained under weak strategic complementarities between countries are a lot more

optimistic than those observed still now in the literature (using the same game). In assuming

that the efforts in abatement of a subset of countries are not partially cancelled by the

countries outside, we show that a stable agreement exists and it can set together more than

four countries.

18

Moreover, this coalition improves substantially the situation in view of unilateral policies. In

implementing cooperation, the subgroup of countries also incites the non-signatories to abate

their emissions.

When environmental damages are perceived very seriously, cooperation and its impact are

very low. This phenomenon is explained by the fact that individual emissions already tend

toward the incompressible emission levels. Even if countries would decide to cooperate they

could not abate below this threshold.

With respect to the previous models, free riding is considered as the most important hindrance

to international environmental cooperation. But, in postulating that countries’ strategies are

substitute, this kind of behaviour is over weighting, yielding to a very low level of

cooperation even in the most serious situations.

When countries exhibit strategic complementarities, the free rider incentives always exist in

the sense that it’s not the grand coalition that forms but an intermediate agreement. However,

free-riding takes the form of a less important effort of abatement from non-signatories with

respect to the effort of signatories.

6. Appendix

The proofs of result 1, 2, 4 and 6 can be established by differentiation.

Proof result 1:

Differentiating the payoff function with respect to x and y, we find that countries’ strategies

exhibit complementarities:

0)(4

),( 2/32

>+=∂∂

∂ −yxcyx

yxf

Under our assumptions, the payoff function is concave which imply continuity of the best-

response function of each country. Moreover, the slope of the latter is always less than one on

the feasible strategies’ space:

11

1)(2 <−

=∂

∂γy

yx

Thus, it will cut only once the first axis and the equilibrium emission level for one country is

unique. Finally, in differentiating ncx and ncz with respect to n and γ, we find:

19

0)( 22

2>

−=

∂∂

nnxnc

γγ ;

0)( 22

4>

−=

∂∂

nnznc

γγ ;

0)(

222 <

−−

=∂∂

nnxnc

γγ

γ;

0<∂∂

=∂∂

γγncnc xnz

Similarly, the proof of the last point is established by differentiating ncf with respect to n

and γ.

0)(2)1(

2/322/1

22<

−−

=∂∂

nnb

nf nc

γγγ

0)(

)12(2/32

22/1<

−−+−

=∂∂

nnbnf nc

γγγ

γ

Proof result 2:

In the full cooperative case, we find that:

0<∂∂

nxc ; 0>

∂∂

nzc ; 0<

∂∂γcx ; 0<

∂∂γcz ;

And,

0<∂∂

nfc ; 0<

∂∂γcf .

Proof result 3:

To establish the first point we compute:

0))(1(

)1(22

22<

−−−

=−nn

nxx ncc γγγ

and 0)( <−=− nccncc xxnzz ;

Then, being a signatory or a non-signatory, the damages a country bears is the same. Hence,

ncc ff > is evident from the preceding point;

For the last point we find: 01 >∂∂

nI ; 01 <

∂∂γI ; 02 <

∂∂

nI ; 02 >

∂∂γI .

20

Proof result 4:

Similarly to the first point of the proof of result 3, we compute:

0)(

)1(22

2<

−+−−

=−ssns

snxx nss γ

0))(1)((

)1(22

2>

−−+−−

=−nsns

snxx nsnc γγ

The inequality nss ff < is straightforward.

For the both last points, we find:

0<∂∂

sxs ; 0<

∂∂

sxns ; 0<

∂∂

szT and 0>

∂∂

sf s ; 0>

∂∂

sf ns

21

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